Densitometria I. Discrete groups
Szil\' ard Gy. R\' ev\' esz, Imre Z. Ruzsa
TL;DR
This work develops a robust framework for densities and means on discrete commutative groups, introducing the upper mean with restricted additivity and its lower conjugate, and studying when a mean exists and how to extend it. It identifies extremal means, namely the uppermost mean $M^*$ and the lowest mean $M_*$, as natural bounds governing how a function behaves under averages over translates, and analyzes commarginality as an equivalence that preserves mean values across such means. The paper then defines expansion, prescribed-value, and density constructions, linking densities to means and exploring regularity, tiles, and witnesses. It culminates with a systematic treatment of density concepts (upper/lower) and their connections to packing, covering, and measure-theoretic properties, including a discussion of when densities can be mensural or regular.
Abstract
An upper mean here is a subadditive functional $\overline M$ defined on bounded functions on a commutative group which has, beside some natural requirements, the property we call restricted additivity: if $g(x)= f(x)+f(x+t)$, then $\overline{M} (g)= 2 \overline{M} (f)$. This tries to grasp that it should not depend on local properties. This naturally induces a lower mean, and when they coincide it is the mean. Restriction to 0--1 valued functions (sets) is a density. We answer the following questions: Given a functional defined on a subset of all functions, when is it a mean? Given a functional, which is a mean, how do we find the upper mean it came from? Is it unique? Given a function $f$, what are the possible values of $\overline M(f)$, for upper means $\overline M$? In particular, we find the extremal means and give several expressions for it. We propose the names ``lowest and uppermost mean'' for them to replace the not really justified names ``lower and upper Banach mean and density''. We also consider analogous questions for densities, with partial answers only.